Computing numerical solutions to Hele–Shaw flow (and related moving boundary problems) can be a challenging task, as interfacial patterns develop which then requires solving the governing equations in complicated moving domains. Such approaches can be classified as either front tracking, where the interface is solved for explicitly, or front capturing, where the interface is represented implicitly. For the classic Hele–Shaw problem, as the pressure of the viscous fluid satisfies Laplace's equation, the most popular choice is the boundary integral method (also known as the boundary element method), which is classified as a front tracking method. In particular, the boundary integral method has been used to solve the classic one-phase Hele–Shaw problem [1]}, [2]}, [3]}, as well as two-phase flow [4]}, [5]}, time-dependent plate gap [6]}, [7]}, and Hele–Shaw flow in channel geometry [8]}. However, for non-standard Hele–Shaw configurations, the pressure may no longer be harmonic and the boundary integral method becomes a less suitable tool. Another disadvantage of front tracking methods is that the mesh may need to be regenerated as the interface evolves, in which case care must be taken to avoid mesh distortion effects.
